{"paper":{"title":"On large deviation probabilities for empirical distribution of branching random walks: Schr{\\\"o}der case and B{\\\"o}ttcher case","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Hui He, Xinxin Chen (ICJ)","submitted_at":"2017-04-12T14:47:45Z","abstract_excerpt":"Given a super-critical branching random walk on $\\mathbb{R}$ started from the origin, let $Z\\_n(\\cdot)$ be the counting measure which counts the number of individuals at the $n$-th generation located in a given set. Under some mild conditions, it is known in \\cite{B90} that for any interval $A\\subset \\mathbb{R}$, $\\frac{Z\\_n(\\sqrt{n}A)}{Z\\_n(\\mathbb{R})}$ converges a.s. to $\\nu(A)$,  where $\\nu$ is the standard Gaussian measure. In this work, we investigate the convergence rates of $$\\mathbb{P}\\left(\\frac{Z\\_n(\\sqrt{n}A)}{Z\\_n(\\mathbb{R})}-\\nu(A)>\\Delta\\right),$$ for $\\Delta\\in (0, 1-\\nu(A))$,"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1704.03776","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}